An accident victim with a broken leg is being placed in traction. The patient wears a special boot with a pulley attached to the sole. The foot and boot together have a mass of 4.0 kg, and the doctor has decided to hang a 6.0 kg mass from the rope. The boot is held suspended by the ropes, as shown in FIGURE P6.40, and does not touch the bed. a. Determine the amount of tension in the rope by using Newton's laws to analyze the hanging mass. Hint: If the pulleys are frictionless, which we will assume, the tension in the rope is constant from one end to the other.

Verified step by step guidance

1. Identify the forces acting on the hanging mass. The only force acting on the 6.0 kg mass is its weight, which is given by F = m * g, where m is the mass and g is the acceleration due to gravity (9.8 m/s^2).
2. Calculate the weight of the hanging mass using the formula F = m * g. Here, m = 6.0 kg and g = 9.8 m/s^2.
3. Since the pulleys are frictionless, the tension in the rope is the same throughout. Therefore, the tension in the rope supporting the hanging mass is equal to the weight of the hanging mass.
4. Write the equation for the tension in the rope: T = m * g, where T is the tension, m is the mass of the hanging weight, and g is the acceleration due to gravity.
5. Substitute the values of m and g into the equation to find the tension in the rope.

Key Concepts

Newton's Laws of Motion

Newton's Laws of Motion describe the relationship between the motion of an object and the forces acting on it. The first law states that an object at rest will remain at rest unless acted upon by a net external force. The second law quantifies this relationship, stating that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F=ma). The third law states that for every action, there is an equal and opposite reaction.

Tension in a Rope

Tension is the force transmitted through a rope or string when it is pulled tight by forces acting from opposite ends. In a frictionless pulley system, the tension is constant throughout the rope. This means that the tension acting on the hanging mass is equal to the weight of the mass minus any other forces acting on the system, such as the weight of the boot in this scenario.

Equilibrium of Forces

An object is in equilibrium when the net force acting on it is zero, meaning all forces are balanced. In the context of the problem, the forces acting on the boot and the hanging mass must balance out. This involves considering the gravitational force acting on the masses and the tension in the rope, allowing us to set up equations to solve for unknowns, such as the tension in the rope.